We begin with some general observations about non-negative irreducible matrices.
Let \(M\)  be an irreducible matrix with spectral radius \(r\) .
It is known that if \(M\)  is a strictly positive matrix, then the limit \(\lim _{k\rightarrow \infty } M^k/r^k\)  exists [1]}.
While this limit need not exist in general if \(M\)  is irreducible, using similar arguments, one can show that there are constants \(c_1,c_2>0\)  such that for all \(n\)  sufficiently large, either \(M^n_{i,j}=0\)  or
\(c_1 r^n\le M^n_{i,j}\le c_2 r^n.\) 
